Tool · Playground

Map composer

Build new 1D maps by composing two known ones. Pick F and G, pick the order or blend, then explore the resulting dynamics: cobweb, time series, bifurcation, Lyapunov exponent.

F

f(x) = r·x·(1 − x)

r = 3.950 ∈ [2.5, 4]

G

f(x) = μ · min(x, 1 − x)

μ = 1.920 ∈ [1, 2]

Composition

x₀ = 0.310Iterations = 80

Orange curve: composed map y = h(x). The cobweb walks x → h(x) → diagonal → x, …

Things to try

Logistic ∘ tent: composing a smooth quadratic with a piecewise-linear map gives a non-trivial unimodal landscape with a shifted bifurcation cascade.
Chebyshev ∘ Chebyshev: Chebyshev maps are topologically conjugate to logistic at r=4. Composing them stays chaotic but reshapes the invariant measure.
Linear blend α·F + (1−α)·G: sweep α and watch the bifurcation skeleton morph between two distinct routes to chaos.
G ∘ F vs F ∘ G: they share Lyapunov sign asymptotically but produce different invariant densities. Toggle and compare.

Compare with the simpler logistic playground and the bifurcation tool.

FAQ